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16.A. Interpret the Global Continuity Theorem 16.1 for the real functions f(x) = x? and g(x) = 1/x, x 0. Take various open and closed sets and con- sider their inverse images underf and g.

16.A. Interpret the Global Continuity Theorem 16.1 for the real functions f(x) = x? and g(x) = 1/x, x 0. Take various open and closed sets and con- sider their inverse images underf and g.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Help with Exercise 16.A.

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16.A. Interpret the Global Continuity Theorem 16.1 for the real functions f(x) = x and g(x) = 1/x, x# 0. Take various open and closed sets and con- sider their inverse images under f and g. 16.B. Let h be defined on R by h(x) 1, 0 < * < 1, 0, otherwise. Exhibit an open set G such that h-1(G) is not open, and a closed set F such that h-(F) is not closed.
Transcribed Image Text:16.A. Interpret the Global Continuity Theorem 16.1 for the real functions f(x) = x and g(x) = 1/x, x# 0. Take various open and closed sets and con- sider their inverse images under f and g. 16.B. Let h be defined on R by h(x) 1, 0 < * < 1, 0, otherwise. Exhibit an open set G such that h-1(G) is not open, and a closed set F such that h-(F) is not closed.
16.1 GLOBAL CONTINUITY THEOREM. The following statements are equivalent: (a) f is continuous on its domain D. (b) If G is any open set in R°, then there exists an open set Gi in Rº such that G,n D = ƒ'(G). (c) If H is any closed set in Rº, then there exists a closed set H1 in RP such that HiD = f-1(H).
Transcribed Image Text:16.1 GLOBAL CONTINUITY THEOREM. The following statements are equivalent: (a) f is continuous on its domain D. (b) If G is any open set in R°, then there exists an open set Gi in Rº such that G,n D = ƒ'(G). (c) If H is any closed set in Rº, then there exists a closed set H1 in RP such that HiD = f-1(H).
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